Stochastic Screen Halftoning

JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION

Vol. 8, No. 4, December, pp. 423–440, 1997ARTICLE NO. VC970363

Stochastic Screen Halftoning for Electronic Imaging Devices

Qing Yu and Kevin J. Parker

Department of Electrical Engineering, University of Rochester, Rochester, New York 14627E-mail: [email protected]

Received June 12, 1997; revised September 12, 1997

to the development and use of stochastic screens for digitalimaging applications.For numerous digital imaging applications, there is a need

to maintain the highest quality perceived image, while utilizinga printer or display that can only achieve a limited number of 1.1. Ordered Ditheroutput states. Digital halftoning is the approach that has been

The history of halftoning technology can be dated backwidely used to meet this demand. In this focus paper, we provideto the last century when physical screens and gauzes werea short summary of halftone techniques, then we concentrate

on the newer and expanding roles of stochastic halftone used to generate halftone images. These techniques havescreens—which are free of regular periodic structures and have been translated directly to digital halftoning. Some excel-numerous advantages in quality color rendering. We address lent comprehensive reviews have been published, includingsome theoretical issues, design and optimality issues, printer Ulichney [1], Roetling and Loce [2], Jones [3], and Kangcompensation issues, and color quality issues that pertain to [4]. A brief orientation is given here. Ordered dither is thethe development and use of stochastic screens for electronic natural digital solution, where a two-dimensional thresholdimaging devices. 1997 Academic Press

array is designed and the halftoning process is accom-plished by a simple pixelwise comparison of the gray scaleimage against the array (Fig. 1). This method is straightfor-

1. INTRODUCTION ward and requires little computation; thus ordered ditheris the most popular and widely used technique. Depending

In numerous digital imaging applications, there is a needon the progressive ordering of how halftone dots in a cell

to maintain the highest quality perceived image, while uti-are turned on/off, ordered dither can be classified into

lizing a printer or display that can only achieve a limitedclustered-dot and dispersed-dot. In clustered-dot ordered

number of output states. Common examples of this includedither, adjacent pixels are turned on as gray level changes

the use of ink-jet printers to make video hardcopy, andto form a cluster in the halftone cell. Clustered-dot dither

the display of images from web pages.is primarily used for printing devices that have difficulty

In the historic evolution of printed images, photographicprinting isolated single pixels. Obviously, this congregation

halftone screens were used to render the illusion of grayof pixels will result in noticeable low-frequency structures

scale with binary (black ink on white paper) printers. Grayin the output image. On the other hand, in dispersed-

regions were printed as a mosaic of black and white subre-dot ordered dither, halftone dots in a cell are turned on

gions, since the properties of the human visual systemindividually without grouping them into clusters. There-

(HVS) would tend to create a perception of gray. In mod-fore, sharp edges can be better rendered compared to clus-

ern digital imaging applications, digital halftone screenstered-dot dither. However, dispersed-dot techniques are

or algorithms are employed to generate precisely definedmore susceptible to dot gain, a problem that is considered

patterns for halftoning. In cases where a limited numberin later sections. Figure 2 gives an example of clustered-

of intermediate states are achievable, the process is some-dot and dispersed-dot patterns.

times referred to as multitoning.In this focus paper, we provide a short summary of 1.2. Stochastic Processes

halftone techniques and concentrate on the newer andexpanding role of stochastic halftone screens—which are Many problems of ordered dither, including susceptibil-

ity to Moire patterns and highly visible texture, can befree of regular periodic structures and have numerous ad-vantages in high quality color rendering. We address some traced back to its rigid regular structures. To break these

regular structures, researchers sought less obtrusive half-theoretical issues, design and optimality issues, printercompensation issues, and color quality issues that pertain tone patterns. Blue noise halftoning, also called stochastic

4231047-3203/97 $25.00

Copyright 1997 by Academic PressAll rights of reproduction in any form reserved.

424 YU AND PARKER

FIG. 3. Flowchart for standard error diffusion.

pending on the input image value. It forces average tonecontent to remain the same and attempts to localize thedistribution of tone levels. Figure 3 shows the flowchartfor error diffusion. This approach was first presented byFloyd and Steinberg back in the 1970’s [5]. Subsequently,many modifications and derivations have been proposedin the design of error filter [8], threshold value [9], feedbackloop [10], as well as processing sequence [11]. Although all

FIG. 1. Ordered dither halftoning technique. these algorithms require intensive computation and someartifacts exist, the quality of the halftone image, particu-larly the sharp edges and many image details, is generallyconsidered excellent [12]. The success of error diffusionscreening or frequency modulated (FM) screening, has

been the most active research field in digital halftoning in lies in the fact that it is a ‘‘good blue-noise generator,’’ aspointed out by Ulichney [1]. In the academic literature,recent years. These terms have been loosely applied to

both algorithm approaches and the screen approach. Error the nature of noise is often described by a color name; i.e.,white noise is so named because of its flat power spectrum.diffusion [5] is the algorithm approach that has been most

extensively studied, whereas the Blue Noise Mask (BNM) Blue noise, on the other hand, has most of its energylocated at high spatial frequencies with very little low-[6, 7] is the term first applied to a screen or threshold array

that produces unstructured, visually appealing halftone frequency component. A typical blue-noise radial averagepower spectrum (RAPS) is shown in Fig. 4. Patterns withpatterns. In order to follow a precise definition from now

on, the term ‘‘stochastic screening’’ applies to a threshold blue-noise characteristics generally enjoy the benefits ofaperiodic uncorrelated dot patterns without low-fre-array. Also, ‘‘mask’’ and ‘‘screen’’ will be used interchange-

ably when both will refer to a threshold array. quency graininess.

1.2.2. Stochastic Screen1.2.1. Error Diffusion

Error diffusion is an adaptive algorithm that produces Stochastic screen halftoning is the subject of active re-search. It combines the simplicity of ordered dither with thepatterns with different spatial frequency content de-

FIG. 2. Halftoned image from clustered-dot halftoning (left) and dispersed-dot halftoning (right).

STOCHASTIC SCREEN HALFTONING 425

sation and screens for multilevel-output devices. In Section6, color halftoning is investigated, and different stochasticcolor halftone schemes are presented, followed by an eval-uation based on a human visual model. Finally in Section7, a summary is given, and current problems with stochasticscreen halftoning are identified and future research is pro-posed.

2. THE CONSTRUCTION OF A STOCHASTICSCREEN—BLUE NOISE MASK

FIG. 4. A blue-noise radial average power spectrum. In this section, the algorithm [6, 7, 13, 14] to generatea Blue Noise Mask is presented. First, an initial blue-noisebinary pattern b[i, j, g] (two-dimensional binary pattern atgray level g) for some intermediate level g (0 , g , 255,blue-noise quality of error diffusion (see Fig. 5). Stochastic

screen halftoning is a point comparison process, so it is assuming an 8-bit mask) is required. Using the filteringand swapping technique presented in Section 3, such aeasy to implement. Thus, devices currently using ordered

dither technique may be switched to stochastic screen half- pattern with a blue-noise characteristics is obtained andused as the initial pattern. From this initial pattern, antoning simply by replacing the original dither array with

a stochastic screen. The halftone image from a stochastic initial mask m[i, j] is generated, which when used to half-tone the constant gray image of level g, produces the initialscreen will have the typical visually pleasing blue-noise

characteristics, which is guaranteed when screens are gen- binary pattern b[i, j, g].Once level g is completed, level g 1 1 is processed (Fig.erated from blue-noise dot patterns of individual gray lev-

els. The Blue Noise Mask, proposed by Mitsa and Parker, 6). For this level, the blue-noise pattern is created by con-verting the appropriate number (the total number of pixelswas the first stochastic screen to realize the above scheme

[6, 7, 13]. in the binary pattern divide by the total number of levels)of 0’s to 1’s in the previous pattern g. At the same time,The following sections will concentrate on the design of

stochastic screens and their applications in black-and-white the mask m[i, j] is updated. This process is repeated untilthe mask has been updated for all the levels above g tohalftoning, multitoning, and color halftoning. Our review

focuses on the scientific literature published in peer-re- level 255. Analogous procedures are used to construct themask for all the levels below g to level 0. The resultingviewed forums. The organization is the following: In Sec-

tion 2, the construction of the prototypical stochastic two-dimensional array m[i, j] will be the final Blue NoiseMask (Fig. 7).screen, the Blue Noise Mask, is outlined. Section 3 details

the common filter approaches in screen construction, and There is a significant constraint on the converting andswapping operation in this mask construction. In makingvarious filter design techniques are examined. In Section

4, the optimality of blue-noise binary pattern in terms of a mask, the binary patterns at different levels are depen-dent. For example, in the upward construction process, allscreen design is pursued. In Section 5, various modifica-

tions of stochastic screens in order to meet special applica- the 1’s in the binary pattern for level g are contained inthe binary pattern g 1 1, so when converting and swappingtions are introduced, such as screens with dot-gain compen-

FIG. 5. Halftoned image from error diffusion (left) and Blue Noise Mask (right).

426 YU AND PARKER

where hhpg(i, j) is a high pass filter designed for level g,

bg(i, j) is the binary pattern, and ‘‘**’’ denotes convolu-tion with circular ‘‘wrap-around’’ properties. In the trans-form (or spatial frequency) domain:

E(k, l) 5 B(k, l)[Hhpg(k, l) 2 1]. (2)

Since Hhpg(k, l) is chosen as a blue-noise (highpass) filter,

then the overall filter [Hhpg(k, l) 2 1] is a lowpass one.

Thus, an essential feature of this filtering is that binarypattern ‘‘clumps’’ in the image domain (corresponding tolow-frequency energy in the transform domain) can belocated by directly filtering the binary pattern. Also, thehighpass region (or lowpass region) can be related to theprincipal frequency fg [1], which is a function of the graylevel g. Mitsa and Parker also suggested that the filterFIG. 6. Blue Noise Mask construction: from level g to g 1 1.Hhpg

could be made adaptive to directly shape the RAPSof the binary pattern for level g. That is,

Hhpg(r) 5 ÏD(r)/Bg(r), (3)1’s and 0’s, these common 1’s shared by the two neigh-

boring levels cannot be changed.where D(r) is the desired blue-noise RAPS for levelThe construction technique outlined above is quite gen-g 1 1 and Bg(r) is the known RAPS for level g. As with anyeral and has enabled the generation of masks with differentfiltering approach, care should be taken to avoid unwantedproperties such as 8-bit depth (level 0–255) and 12-bitdiscontinuities in the transform domain that produce ‘‘ring-depth (level 0–4095), small size (64 by 64) and large sizeing’’ in the image domain. This approach is depicted in(256 by 256), isotropic and anisotropic.Figs. 8 and 9. Note that this specific filter is computationallyThere are two critical parts in designing a Blue Noisemore involved than simple lowpass filters, but the approachMask: the digital filters and the optimality issue. Thesedemonstrates the central requirement of providing a low-topics will be covered in Sections 3 and 4, respectively.pass filter with a cutoff frequency linked to gray level gby the principal frequency fg [15]:3. COMMON FILTER APPROACHES

As described in the previous section, the algorithm forgenerating the Blue Noise Mask recognizes that in a half-tone array, the binary pattern at any gray level g 1 1 canbe thought of as being ‘‘built up’’ from the binary pattern atlevel g. And furthermore, the Blue Noise Mask algorithmutilizes the concept of filtering a binary pattern from levelg to select the location of pixels that would be the ‘‘bestcandidates’’ for addition of majority pixels required forlevel g 1 1. Once binary patterns for all gray levels havebeen sequentially produced in this way from some ‘‘seed’’level, the binary patterns can be summed to produce athreshold array or Blue Noise Mask. The role of filters inthis procedure warrants close attention, and a summary ofsome important results is given below.

Mitsa and Parker [6, 7, 13] selected the location of pixelsthat should be changed from minority to majority valuesby finding the extremes of an error function e[i, j]. Thiserror function was generated by directly filtering the binarypattern of level g and then subtracting the binary patternfrom the filtered pattern. Specifically, in the image domain

FIG. 7. Adding up dot patterns of each gray level to make a Bluee(i, j) 5 [hhpg

(i, j) p p bg(i, j)] 2 bg(i, j), (1) Noise Mask.


FIG. 8. A desired blue-noise RAPS D(r) (solid line) and an FIG. 10. Principal frequency (solid line) and cutoff frequency ofactual blue-noise RAPS Bg(r) (dotted line) at gray level 210. the highpass filter (dashed line) for each gray level.

(or Gaussian width parameter) with gray level [14, 16].fg 5HÏg, for g # 1/2

Ï1 2 g for g . 1/2,(4) Yao and Parker [14] also demonstrated that a variety of

lowpass filter shapes could produce desirable halftone pat-terns, so long as the filter parameters were adjusted forwhere g is the gray level normalized to 1. This relationshipappropriate cutoff with respect to the principal frequency.is plotted as the solid line in Fig. 10. The dashed line inMitsa and Brothwarte [19] further developed the conceptFig. 10 corresponds to the factor of 1/Ï2 (in principalof a filter bank (for different gray levels), where lowpassfrequency, or Ï2 in average separation) that was discussedfilters were each adjusted for a specific cutoff below theby Mitsa and Parker [16] as an empirical choice of transi-principal frequency (K 5 As), and this was related to ation cutoff frequency for some filters. In a later paper,wavelet-type filter bank. Dalton [20] described the use ofParker, Mitsa, and Ulichney [17] demonstrated how, at abandpass filters to produce textured binary patterns.single gray level, changes in the filter cutoff frequency

Thus, the general lessons from these works are thatcould produce different types of final halftone patterns.direct filtering of binary patterns can be useful in selectingSpecifically, for fc representing the cutoff frequency of apixels that can be changed to produce a desired resultfilter, let fc 5 Kfg , where K is an adjustable scaling factor.in the image domain, and correspondingly approximateWhile K varies from 0.5 to 1.0, binary patterns of differenta desired power spectrum in the transform domain. To‘‘textures’’ were produced, with K 5 1/Ï2 employed forillustrate the varieties of filters that have been used, Figs.general use.11 and 12 depict the transform domain and correspondingUlichney [18] further explored the filter issue, choosing aimage domain filters from Mitsa and Parker [6, 16], Ulich-two-dimensional Gaussian filter implemented in the imageney [18], and Yao and Parker [14], in each case the filterdomain for direct operation on the binary pattern. Herepresented is the one that would be used for gray level 210demonstrated that, for small size arrays (less than 32 byout of 256. Only isotropic filters are shown, but anisotropic32, for example), even a single Gaussian filter that is notfilters have also been used [14].adjusted for gray levels (as in the previous work) could

Note that the specification of a filter as a lowpass filterproduce a useful Blue Noise Mask for some applications.in the image domain provides perhaps the easiest wayOf course, the seed pattern and Gaussian filter width re-to explain the algorithm to new designers. However, thequire careful choice in order to produce a desirable result.benefit of specifying the filter in the transform domain and,In general, it is beneficial to vary the cutoff frequencyas an initial highpass operation, is that the final RAPS of

FIG. 9. Overall lowpass filter [Hhpg(k, l) 2 1] and original highpass

filter Hhpg(k, l). FIG. 11. Different filters in frequency domain.

428 YU AND PARKER

FIG. 12. Different filters in spatial domain. FIG. 14. Two filters in frequency domain. A small ‘‘dip’’ is seennear frequency sample 110 in one curve.

the binary pattern will generally approximate the shape ofthe highpass filter that is specified in Eq. (2) or (3) [14, 16, 1. Gaussian,17]. Thus, a halftone designer considering the final RAPSof the binary pattern can envision changes resulting from

F(u, v) 5 e2(u21v2)/2s2

; (5)different filter shapes by considering the filter as an initialhighpass filter in transform domain as taught in the early

2. Gaussian with a band-reject component,references [6, 7, 13, 16].Modifications to the filters may be further explored. One

F(u, v) 5 e2(u21v2)/2s2

2 ae2((Ïu21v22fg)2)/2s92. (6)interesting filter application is the manipulation of the spec-

tral peak at the principal frequency. In filtering the binaryFigure 14 shows the two filters in transform domain,pattern with a simple lowpass filter, such as a Gaussian,

where the parameter ‘‘a’’ of Eq. (6) is set as 0.02 for thethe low frequencies are minimized by the algorithm whichsecond filter. Each of these two filters was repeatedly ap-changes certain identified pixels [14]. By careful choice ofplied to the starting pattern with pixel swapping until thea bandpass filter, one can enhance selective frequencies inperceived mean square error (MSE) between the binarythe transform domain, through the proper identificationpattern and its corresponding uniform gray pattern stoppedof certain pixels in the image domain.decreasing. The resulting patterns are given in the bottomFor example, consider a starting white-noise binary pat-part of Fig. 13 and the corresponding RAPS in Fig. 15.tern at level 210 shown in Fig. 13. Now apply the following

Thus, selective enhancement (or suppression) of regionstwo different filters individually to that pattern:of the power spectrum can be designed by proper selectionof filters. Note also that the difference in lowpass filters(Fig. 14) is very subtle but the resulting binary patternshave a recognizably different RAPS. The examples givenhere demonstrate the utility in designing binary patternswith particular characteristics.

The properties of the human visual system can also beincorporated into the filter. Important research combiningthe power spectrum and human visual system (HVS) con-cepts with binary patterns was reported by Sullivan, Ray,and Miller [10]. Utilizing a model of the low-contrast phot-

FIG. 13. Upper: white noise starting pattern; Left: final patternfrom filtering and swapping with a Gaussian filter; Right: final patternfrom filtering and swapping with a Gaussian filter which has a band-reject component. FIG. 15. RAPS for the two final blue-noise patterns.


respect to the original uniform gray pattern. The analysis ofthe filtering technique put a lower bound on the achievableperceived MSE, assuming that a filter based on the humanvisual system is also used to measure the perceived MSEbetween the gray and binary patterns. As Yao pointedout, the difference between the local filtered output of thelargest white clump and the largest black clump must begreater than a certain value T in order for the perceivedMSE to be further reduced. T is given by

T 5 1/4ls 2, (8)

where s is the standard deviation of a Gaussian filter basedon a human visual model.

FIG. 16. Spatial Frequency Response of the human visual model In another way of speaking, a nonzero lower limit onby Sullivan et al. [10]. the perceived MSE will be reached when the filtering tech-

nique is employed.To exceed this limit, a postfiltering algorithm [22] is

introduced below. By locally enforcing a vector processopic modulation transfer function as illustrated in Fig. 16,after filtering, perceived MSE is further reduced and morethey were able to generate a locally unstructured ‘‘tileable’’visually pleasing binary patterns are obtained. This newbinary pattern, 32 by 32 square, for each gray level, andalgorithm will be presented first. Then, a series of binarythey used a cost function with HVS weighting to guide apatterns for a certain gray level are generated varying fromMonte-Carlo approach with simulated annealing in thea white noise pattern to a highly structured pattern. Bycreation of individual binary patterns. As a general rule,studying these patterns in both image and transform do-the human eye has maximum sensitivity in the horizontalmain, the optimality issue is investigated.and vertical directions and minimum sensitivity in the diag-

onal direction. Therefore, an anisotropic filter may be de-4.1. Electrostatic Force Algorithmsigned so that the resulting pattern will have more energy

in the diagonal direction than in the horizontal and vertical This force algorithm is based on the model of electro-directions. For example, an original two-dimensional static force between electric charges. Point charges of sameGaussian filter F(u, v) may be modified in the following polarity repel each other while charges of different polarityway: attract. The force is proportional to 1/r2, where r is the

distance between the two charges. Therefore, in case of aF9(u, v) 5 [1 1 c p cos(4u)] p F(u, v), (7) binary pattern whose pixel values are either 1 or 0, if all

the majority pixels are treated as point charges with ‘‘1’’polarity and all the minority pixels are treated as pointwhere u is the central angle around the DC point and c ischarges with ‘‘2’’ polarity, then there will be interactivea constant between 0 and 1 which controls the amount offorce between all the pixels. Furthermore, if those minorityenergy distribution.pixels are allowed to move freely under the net electrostaticforce from pixel charges in a certain neighborhood W,4. OPTIMALITY OF BLUE-NOISE BINARY PATTERNafter some number of iterations, a nearly homogeneousAND STOCHASTIC SCREENdistribution of those minority pixels should be expected.Since every pixel has some force acting on it, a thresholdThe filtering techniques presented in Section 3 produce

visually pleasing blue-noise binary patterns. However, are value should be set such that only when the net force ona pixel surpasses the threshold value, a pixel move in thethese binary patterns also optimal for their corresponding

gray levels and for the construction of a mask? These direction which would minimize the net forces on thatminority ‘‘charge.’’ Remember that the starting pattern forquestions will be addressed in this section.

Yao [21] has given a detailed mathematical analysis of this force algorithm is the one we obtain from the filteringprocess, which is already free of clumps. Therefore, theBlue Noise Mask construction based on a human visual

model (similar to the model used by Sullivan et al.), which histogram of the net force (either in horizontal or verticaldirection) on every minority pixel will be highly peakedprovides insights to the filtering process and also prescribes

the locations of the dots that, when swapped, will result near 0. In this case, it is reasonable to assume the meanvalue of net force on every minority pixel to be 0, andin a binary pattern with minimum perceived MSE with

430 YU AND PARKER

FIG. 17. Perceived MSE vs iteration in filtering and swapping process.

the threshold value (TH) can be related to the standard shows the difference between the largest white clump andthe largest black clump (DWB) for each iteration. Sincedeviation (SD) of the net force asthe initial pattern is a white-noise one, the DWB is quite

TH 5 V · SD. (9) large and the perceived MSE decreases in each iteration.After a certain number of iterations, the DWB approachesV is a variable that is adaptive to the gray level as well asthe lower bound T set in Eq. (8) (in this case approximatelythe iteration number. As binary patterns are two-dimen-0.0137), then the filtering process can no longer improvesional, the force calculation and pixel movement must bethe binary pattern (Fig. 19). Figure 20 shows the binarydone in the horizontal and vertical directions, respectively.pattern (P2) obtained from the filtering process with per-

A different force-relaxation model for adaptive halfton- ceived MSE of 0.263.ing of images was proposed by Eschbach and Hauk [23]. From this pattern (P2), the force algorithm is carried

out. The neighborhood W, which is used to calculate the4.2. A Progressive Series of Binary Patternsnet force on each pixel, is set as 13 by 13 and the starting

To illustrate the previous procedure, a white-noise pat- value of V is set as 1.5. Figure 20 shows the binary pat-tern at level 245 is used as the initial pattern, then the tern (P3) after just five iterations with perceived MSE

of 0.165.transform-domain filtering is applied Fig. 17 shows theperceived MSE drop versus iteration number and Fig. 18 It is quite obvious that by locally enforcing the vector

FIG. 18. Difference between the largest white clump and the largest black clump of filter output vs. iteration in the filtering andswapping process.


FIG. 21. RAPS of patterns P1, P2, P3, and P4.

from a ‘‘seed’’ pattern, the choice of ‘‘seed’’ pattern shouldbe based on its suitability for mask generation. In anotherFIG. 19. Electrostatic force model.way of speaking, an optimal binary pattern should notdegrade the quality of its neighbor levels (g 1 1 andg 2 1 and so forth for one level g).force process, the perceived MSE are further reduced and

Obviously, a white-noise pattern cannot be optimal.more uniform patterns are generated.However, the highly structured pattern is not optimal ei-ther. If this highly structured pattern is used as an initial4.3. Optimality Issuepattern and neighboring levels are constructed, those

Without strict proof here, it is noted that the force algo- neighboring binary patterns are generally visually an-rithm does converge after further iterations. Figure 20 noying due to noticeable disruption of the semi-regularshows the final pattern (P4) obtained when the force algo- patterns established by the specific initial pattern. Thisrithm converges after 75 iterations. The perceived MSE of leads to the question: What pattern between white noiseP4 is 0.087 and this pattern has a highly ordered structure. and highly structured patterns constitutes an optimal blue-

Since all the binary patterns in a mask are constructed noise ‘‘seed’’ pattern for Blue Noise Mask construction?Figure 21 shows the RAPS of all patterns presented in

Figure 20. P1 has the typical white-noise characteristics,P2 and P3 have the typical blue-noise characteristics, andP4 has a very high peak at the principal frequency withan emerging ‘‘second harmonic’’ peak. The trend is veryobvious that spectrum starts from white-noise shape, grad-ually switches to blue-noise shape, and ends up in a shapewith energy concentrated around the principal frequency.Therefore, in carrying out the force algorithm, a criterionshould be set that will enable the computer to terminatethe process once an excessive concentration of energy atprincipal frequency emerges.

Another experiment has been carried out as follows.For an intermediate gray level (level 245), another seriesof binary patterns are generated with the force algorithm.In this case, pattern 0 corresponds to the output fromthe filtering technique, pattern 5 is the output after fiveiterations of the force algorithm on pattern 1 and pattern50 is the output after 50 iterations of the force algorithmon pattern 1. Then for each of these three patterns, itsneighboring gray levels are constructed. Using the HVSmodel by Sullivan et al. (Section 3), a plot of the perceivedMSE between the binary pattern and corresponding uni-

FIG. 20. Patterns P1 to P4 (clockwise from upper left corner). form gray pattern for each gray level is generated, respec-

432 YU AND PARKER

FIG. 22. MSE vs gray level (partial) for Blue Noise Masks con-structed from three ‘‘seed’’ patterns of different blue-noise RAPScharacteristics.

FIG. 24. Printer characteristic curve and lookup table curve forcompensation.tively. These plots are shown in Fig. 22. The plots for

pattern 0 and pattern 50 show very large discontinuitiesaround the initial level (245), which means that they are formed using lookup tables before halftoning. Withnot optimal for mask construction. Therefore, the smooth- stochastic screen halftoning, dot-gain compensation canness of the perceived MSE transition could serve as a be actually included in the screen design process [24]. Inparameter to design an optimal binary pattern. general, these approaches can be classified into two catego-

ries. One is by printing fewer black dots than required in5. SPECIAL APPLICATIONS ideal case, and the other is by printing black dots in a

preferred way while keeping the number of black dots forSo far, all the discussions have considered the design ofeach level untouched.an ideal stochastic screen for an ideal device. However,

since real printers and displays are not ideal, a special 5.1.1. Printing Fewer Black Dotsscreen can be designed to meet individual application re-

The nonlinearity of a specific printer can be directlyquirement. Although these requirements could be met withaccounted for in the construction of a mask. Gray patchesdifferent pre/post processing techniques, by incorporatingof certain levels are first printed to get the printer input-the device characteristics into screen design, both render-output characteristics curve. Then, a corresponding curveing time and memory can be reduced.to compensate for this nonlinearity is generated. This curvewill show how many dots are actually needed to correctly5.1. Dot-Gain Compensationrender a gray level. Thus, instead of converting a pre-set

In digital printing, one major concern is dot gain, which number pairs of 1’s and 0’s to move up/down one level, acan be attributed to ink spread or dot overlap and usually variable number pair of dots are converted according tois a combination of both (Fig. 23). With stochastic screens, the compensation curve. Figure 24 shows the printer char-isolated and dispersed halftone dots are typically gener- acteristic curve, an ideal (linear) mask curve and a lookupated, and therefore dot-gain compensation will be neces- table curve for a printer.

Sometimes, if the printer characteristic curve is not avail-sary. In practice, dot-gain compensation usually is per-able during mask design or if several masks have to bedesigned, a mask with 12-bit depth can be built first insteadof the typical 8-bit ones. The mask construction is exactlythe same as in Section 2, except that 4096 gray levels areconsidered. Once the 12-bit mask is completed, it couldbe easily mapped back to an 8-bit mask when the printercurve is available. Also, with different mapping strategies,many different 8-bit masks might be generated from this12-bit one in a short period.

5.1.2. Printing Dots in a Preferred Way

The two methods presented above try to reduce dot gainFIG. 23. Dot gain model. by printing fewer black dots than in an ideal case, i.e., the


number of black dots printed corresponds to a lighter levelthan the desired one, but the desired level is achieved dueto ink spread or dot overlap. Another way to look at dotgain is that it is related to the area-to-perimeter ratio ofprinted dots. The area of paper covered by a dot is mea-sured in pixels, while the perimeter is the total length oftravel around the outside of a printed dot. It is easy to seethat the smaller this ratio, the bigger the dot gain. For anisolated dot, this ratio is 0.25 assuming each dot has a unitdiameter. In clustered-dot dither where the halftone dotsin a cell are connected, this ratio is bigger than 0.25, whichis the reason that clustered-dot dither generally shows lessdot gain. Thus, another approach to reduce dot gain is toincrease the area-to-perimeter ratio.

The nonsymmetric mask. Generally speaking, dot gaincan be severe for dark gray levels since black dots are themajority. In the construction of a Blue Noise Mask, certainwhite dots have to be replaced with black ones to go fromlevel g to level g-1. Normally, with a lowpass filter pickingup those white dot candidates (as specified in Section 2),connected white dots are more likely to be selected thanisolated white ones. Therefore, as the total number of whitedots is decreasing, the number of isolated white dots is

FIG. 25. Partial gray ramps halftoned with a symmetric maskactually increasing, which leads to a decreased area-to-(top) and a nonsymmetric mask (bottom).perimeter ratio and a potential dot-gain problem. To avoid

this, isolated white dots can be eliminated first as we moveto lower levels, while keeping the connected white dots

5.2. Multitoningintact until all isolated white dots are removed. With thismodified algorithm, dot gain could be reduced to a certain Stochastic screen halftoning could easily be generalizeddegree. Because the resulting binary patterns at darker for use with devices having multilevel output. Typically,levels contain connected white dots in small ‘‘clusters,’’ such techniques are referred to as multitoning. Figure 27the use of this nonsymmetric mask is analogous to automat- shows a generalization of the stochastic screen techniqueically switching the printer to coarser resolution in dark for application to multilevel devices. It can be seen thatregions where dot gain is a problem, as illustrated in Fig. 25. this is equivalent to the binary implementation, except that

a quantization operation replaces the threshold operation.The checkmask. In making this mask, a mid-grayAssume an input image I(x, y) has p different levels andcheckerboard pattern is generated first where each pixel

the output device has q possible levels, also assume theis replicated to a 2 by 2 dot. This replicated checkerboardmask M(x, y) is 256 by 256 in size and has 256 levels. Then,is used as the binary pattern for level 128, hence the namethe multitoning process can be given by‘‘checkmask’’ for the final mask built from this initial pat-

tern. There are two reasons to use a replicated checker-board as a starting pattern. First, the area-to-perimeterratio is increased, compared to a regular blue-noise pattern.Second, it will be suitable to certain printers having diffi-culty at the highest spatial frequency, i.e., those printerswhose modulation transfer functions (MTF) have low cut-off frequency. For an 8-bit mask, a binary pattern at level128 will have its principal frequency at the longest spatialfrequency and most of its energy is actually concentratedaround this frequency. A replicated 2 by 2 checkerboardhas its energy shifted to lower frequencies, thus avoidingthe highest spatial frequency and easing the requirementon a printer. A partial gray ramp of a checkmask is shown

FIG. 26. Partial gray ramp halftoned with a checkmask.in Fig. 26.

434 YU AND PARKER

FIG. 27. Multitoning flowchart.

6.1. Conventional Color Halftoning Approach

In conventional halftoning, the same clustered-dotO(x, y) 5 INT 3I(x, y) 1M(xm , ym)

256p 2 1q 2 1

p 2 1q 2 1

4, (10)screen can be used to halftone the C, M, Y, K planesseparately to obtain four halftone images, which are thenused to control the placing of color on paper. One immedi-ate problem of this scheme is the appearance of Moire

where O(x, y) is the output value, xm 5 x MOD 256 and patterns, which are caused by the low-frequency compo-ym 5 y MOD 256, INT( ) indicates an integer truncation, nents of the interference of different color planes. Whenand MOD stands for the modulation operation. combining periodic signals, such as two color halftone

It can be seen that the screen value is scaled and screens of vector frequency f1 and f2 , interference producesadded to the input image before a simple threshold a ‘‘beat’’ at the vector difference frequency fb 5 f1 2 f2operation. [2]. If the individual color screens were made at the same

Although any stochastic screen designed for black-and- angle and frequency, any slight spatial frequency modula-white halftoning could be used with the above implementation due to misregistration in the printing process forms ation for multitoning, an improvement could be made when low-frequency visually objectionable beat. To reducescreens are optimized specifically for multitoning. Spauld- Moire patterns, the screens are typically oriented at differ-ing and Ray [25] have investigated methods that minimize ent angles, usually 308 apart. At 308-separation, the fre-a visual cost function within a quantization interval; they quency of the Moire is about half the screen frequency,have also reported using nonuniform quantization func- thereby producing a high-frequency ‘‘rosette’’-shaped beattions in the multitoning implementation. pattern. Color errors caused by misregistration only appear

at high frequencies and therefore are not easily detected.To achieve the highest frequency and least visible rosette6. COLOR HALFTONINGpatterns, it is typical practice to orient cyan at 1058, ma-genta at 758, yellow at 08, and black at 458 (Fig. 28). BecauseColor imaging normally requires mixing of three addi-yellow and black have the least and most impact on visualtive primary colors (RGB) for CRT display or subtractivesensitivity, respectively, they are oriented at angles whereprimary colors (CMY) for print. Printing technology canhuman eyes are most and least sensitive. Although yellowalso utilize a fourth primary (K) to provide a betteris at 158 relative to the other color planes, its impact onblack hue, enlarge the color gamut and improve imageintercolor Moire is not objectionable because of low con-quality. Additional colors can be added to further enlargetrast. One recent direction of research on color printing isthe color gamut. Color halftoning is the process ofthe introduction of more colors to expand the color gamutgenerating halftone images for the different color planes[4]. Rotation angles of these colors must be assigned tofor a printing or display device. Color image halftoningreduce Moire patterns. However, there is a limit to theis significantly more complicated than halftoning for anumber of angle selections. Therefore, it can be difficultgray scale image. All the qualities required of black-to apply the conventional color halftoning technique toand-white halftone images apply to color halftone imageshigh-fidelity color printing.that are composed of multiple color planes, but, in

Stochastic halftoning, such as error diffusion and sto-addition, the interactions between color planes must beprecisely controlled. chastic screens, eliminates the Moire concern completely,


This concept of utilizing the finest possible patterns alsoserves as a fundamental rule for designing schemes in usingstochastic screens for color halftoning.

6.3. Color Halftoning Using Stochastic Screens

The following schemes have been proposed to apply theBlue Noise Masks to halftone color images [28].

6.3.1. The Dot-on-Dot Scheme

The simplest application of the mask utilizes the samemask for each color plane; this is known as the dot-on-dottechnique. Although this approach is easiest to implement,it is rarely used in practice because it results in the highestlevel of luminance modulation and the output is most sensi-tive to misregistration.

Suppose a gray patch is to be rendered with equal levelsof cyan, magenta, yellow. With this dot-on-dot approach,

FIG. 28. Screen angle rotation in conventional color halftoning.the halftone images for each color plane are identical.Therefore, whenever a cyan dot is printed, a magneta dotand a yellow dot also are printed. As a result, the finaloutput patch will be formed with dispersed black dotssince the halftone dots created by these processes are rela-on the white background. This produces a larger level oftively unstructured. This removes the constraints of theluminance modulation than other cases where color dotsrotation angle. Therefore, high-fidelity color printing isare not coincident, resulting in more visible halftone pat-more realizable with stochastic halftoning.terns. Another possible outcome under the same scenario

6.2. Color Halftoning Using Error Diffusion is that, if one color plane is misregistered relative to others,the annoying color ‘‘banding’’ effect will appear with possi-One characteristic of error diffusion is that it producesble color shifts.correlated dot patterns. Therefore, if error diffusion is ap-

plied to the individual color planes (so-called ‘‘scalar error 6.3.2. The Shifted Mask Schemediffusion’’), the halftone images for different color planes

To decrease correlation of the color planes, spatiallywill be highly correlated. To improve the visual quality ofshifted masks are employed for halftoning each colorcolor halftoning using error diffusion, Miller and Sullivanplane. This will also increase the spatial frequency of the[26] process the color image in a vector color space withprinted dots. For example, one mask can be used for theeach image pixel treated as a color vector. This techniquecyan plane; then it can be shifted in the horizontal andis called ‘‘vector error diffusion.’’ The color image is firstvertical directions in a wrap-around manner and appliedconverted to a nonseparable color space, and each pixelto the magenta plane. Similarly, the mask can be shiftedis assigned with the closest halftone color. The resultingby different amounts and applied to yellow and then tovector halftone error is distributed to neighboring pixelsblack. This technique will tolerate misregistration and,in the same manner as scalar error diffusion.therefore, is more robust for a real printing process. How-Klassen et al. [27] also proposed a vector error diffusionever, if the set of shifts are not chosen carefully, low-technique that minimized the visibility of color halftonefrequency structures may appear when a color pattern isnoise. It is a well-known property of the human visualoverlapped with its shifted version. Generally, the shiftsystem that the contrast sensitivity decreases rapidly withvalues must be tested first by printing some overlappingincreasing spatial frequency. Thus the minimum thresholdgray patches halftoned with the shifted masks and a deci-above which patterns are visible rises rapidly with increas-sion must be made if these shift values are optimal.ing spatial frequency. One approach to increase spatial

frequency, so as to minimize the visibility of color halftone6.3.3. The Inverted Mask Schemenoise, is to select low-contrast color combinations wher-

ever possible and to generate the finest possible ‘‘mosaic’’ In this strategy, one mask is applied to one color planepattern without large clumps. Therefore, light gray is and its inverted version is applied to another. Inverting aprinted using nonoverlapping cyan, magenta, yellow, and mask means taking the 255th complement of a mask:unprinted white pixels, as opposed to printing occasionalblack clusters on a large white background. mi[i, j] 5 255 2 m[i, j]. (11)

436 YU AND PARKER

FIG. 29. A color patch halftoned with different schemes (from left to right, top to bottom: dot-on-dot, shift, invert, four-mask, anderror diffusion).

In light regions, this scheme results in the nonoverlapping 3. Three more binary patterns are made in the sameway by picking the location of pixels with values in thearrangement of color dots with high spatial frequency.

However, this scheme is only applicable for two color range 64–127, 128–191, and 192–255.planes (typically cyan and magenta), and some other 4. This construction ensures that these binary patternsscheme has to be used to determine other color planes. exhibit blue-noise characteristics. The filtering and swap-

ping technique can be further used to eliminate any resid-6.3.4. The Four-Mask Scheme ual periodic structures.

This scheme is actually an extension of the inverted 5. These four binary patterns are used as initial patternstechnique. It is based on the same idea: increasing the to generate four masks.spatial frequency of the printing dots and minimizing the

When these four masks are applied to different colorlow-frequency energy introduced by the overlapping ofplanes, they generate color halftone dots that are maxi-color planes. Unlike the inverted technique, this schememally dispersed, therefore achieving the highest spatialwill not limit the number of masks that can be used, sofrequency, especially at areas of highlight levels.it is more appropriate for generating high-fidelity color

halftone images (Fig. 29). 6.3.5. Evaluation Using a Human Visual Model inIn the case of the four CMYK color planes, four anti- CIELAB Space

correlated masks are generated from four mutually exclu-In this part, a human visual model is used to evaluatesive seed patterns. The steps are given below:

the effects of the four schemes presented above on the1. Generate one Blue Noise Mask that produces visually luminance and chrominance of a color test patch [29].

pleasing unstructured binary patterns. Figure 30 shows the block diagram of this evaluation; theHVS model by Sullivan et al. (Section 3) is used in this2. Assuming that the mask values are from 0 to 255, for

all the mask locations that have a value in the interval 0 case. Both the halftoned and the original color patch arepassed through the human visual model and then convertedto 63, a binary pattern is defined by setting the pixels

corresponding to these locations to black dots, while the to the CIELAB space. The color difference in CIELABspace is given byremaining pixels are set to white.


FIG. 30. Flowchart for color halftone schemes evaluation.

DE 5 Ï2

(DL*)2 1 (Da*)2 1 (Db*)2, (12) dot-on-dot scheme results in minimum chrominance errorbut maximum luminance error and the four-mask schemeresults in minimum luminance error but maximum chromi-where DL*, Da*, Db* are corresponding differences be-nance error, while the result from the shift scheme fallstween two colors. This color difference can be further bro-in between.ken up into components of luminance error DL* and chro-

minance error DC*, which is given by6.3.6. Adaptive Color Halftoning

DC* 5 Ï2

(Da*)2 1 (Db*)2. (13) Beyond the previous methods, one solution to reduceperceived colorimetric error is to apply two mutually exclu-sive masks on two color planes first and then to apply anOur analysis [29] shows that different perceived errors are

produced by different mask techniques. In general, the adaptive scheme on other planes. Another advantage of

FIG. 31. Flowchart of adaptive color halftone scheme.

438 YU AND PARKER

FIG. 32. Details of adaptive decision step.

this adaptive scheme is that color reproduction could be Assuming the viewing distance at 10 inches and printerresolution at 300 dpi, the perceived MSE between thetaken into account [30]. Figures 31 and 32 show the flow-

chart of this new scheme. original image and each color halftone image for the lumi-nance channel and chrominance channel are calculated.For this method, it is necessary to know the L*A*B*

values of the eight primary colors of the destination printer. As the results show [30], the lowest colorimetric error(especially luminance error) is achieved using the adaptiveThus, solid patches of the primary colors are first printed,

then measured, and a lookup table (LUT) of the L*A*B* scheme. The tradeoff is computational complexity. How-ever, by having one color channel be adaptive, increasedvalues for each primary color is generated.

Next, one mask is applied to the cyan plane and its flexibility is obtained to manipulate the output so as toreduce colorimetric error while permitting customizationinverted version on the magenta plane. In this way, the

highest spatial frequency is achieved. At each image pixel, to specific printing hardware. It can be seen that the ap-proach is easily extended to black and other high-fidelitythere are only two possible values for the yellow plane,

either 255 or 0. Therefore, only one from two possible color inks.primary colors (c1 and c2) for that image pixel should beselected. To do that, the corresponding L*A*B* values 7. CONCLUSIONfor c1 and c2 as well as the L*A*B* values for originalpixel (c0) have to be identified. Then the distances in The introduction of stochastic screens in recent years

has opened up new possibilities for rendering images onCIELAB space between c1 and c0 and c2 and c0 are calcu-lated, and the primary color which gives the smaller dis- printing or display devices with limited output states. The

Blue Noise Mask, the prototypical stochastic screen, com-tance is selected. Finally, the luminance and chrominanceerror between the chosen halftone color and original color bines the speed of threshold arrays with the desirable un-

structured patterns of error diffusion. Critical topics forare calculated and passed to neighboring pixels in the errordiffusion sense. analysis and research include the questions of filtering,

optimality, color strategies, and application-dependent de-To compare the performance of this adaptive schemewith other schemes mentioned earlier, a real image is half- sign. These topics have been discussed in the previous

sections. Further research will need to consider refine-toned with all the schemes presented so far and the per-ceived luminance and chrominance errors of the resulting ments to the concept of optimal design for color rendering,

and more advanced models of the human visual perception.halftone images are evaluated [30].


binary patterns,’’ in Proceedings, NIP12: International Conference onREFERENCESDigital Printing Technologies [33, pp. 66–69].

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QING YU received the B.S. degree in Physics (1994), magna cum18. R. Ulichney, The void-and-cluster method for dither array generation,laude, from University of Houston and the M.S. degree in Electricalin Allebach and Rogowitz [32, pp. 332–343].Engineering (1995) from University of Rochester. He is currently a Ph.D.19. T. Mitsa and P. Brathwaite, Wavelets as a tool for the constructioncandidate at the Department of Electrical Engineering, University ofof a halftone screen, in Rogowitz and Allebach, [31, pp. 228–238].Rochester. His research interests include digital printing, color image

20. J. Dalton, Perception of binary texture and the generation of stochas- processing, and human visual system.tic halftone screens, in Rogowitz and Allebach [31, pp. 207–220]. In the summer of 1996, he was employed by Xerox Corporation as a

21. M. Yao, L. Gao, and K. J. Parker, ‘‘An analysis of the blue noise mask software engineer in the Office Document Product Division; in the sum-based on a human visual model,’’ in Proceedings, NIP12: International mer of 1997, he was employed by Eastman Kodak Company as a profes-Conference on Digital Printing Technologies [34]. sional intern in the Image Science Division.

Mr. Yu is a member of IEEE and IS&T.22. Q. Yu, K. J. Parker, and M. Yao, ‘‘Optimality of blue noise mask

440 YU AND PARKER

received in 1978 and 1981. From 1981 to 1985 he was an assistant professorof electrical engineering and radiology. Dr. Parker has received awardsfrom the National Institute of General Medical Sciences (1979), the LillyTeaching Endowment (1982), the IBM Supercomputing Competition(1989), the World Federation of Ultrasound in Medicine and Biology(1991). He is a member of the IEEE Sonics and Ultrasonics SymposiumTechnical Committee and serves as reviewer and consultant for a numberof journals and institutions. He is also a member of the IEEE, the Acousti-cal Society of America, and the American Institute of Ultrasound inMedicine. He has been named a fellow in both the IEEE and the AIUMfor his work in medical imaging. In addition, he was recently named toKEVIN J. PARKER received the B.S. degree in engineering science,the Board of Governors of the AIUM. Dr. Parker’s research interests aresumma cum laude, from SUNY at Buffalo in 1976. Graduate work inin medical imaging, linear and nonlinear acoustics, and digital halftoning.electrical engineering was done at MIT, with M.S. and Ph.D. degrees

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Stochastic Screen Halftoning